Best Known (32−11, 32, s)-Nets in Base 32
(32−11, 32, 6555)-Net over F32 — Constructive and digital
Digital (21, 32, 6555)-net over F32, using
- net defined by OOA [i] based on linear OOA(3232, 6555, F32, 11, 11) (dual of [(6555, 11), 72073, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
(32−11, 32, 20451)-Net over F32 — Digital
Digital (21, 32, 20451)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 20451, F32, 11) (dual of [20451, 20419, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
(32−11, 32, large)-Net in Base 32 — Upper bound on s
There is no (21, 32, large)-net in base 32, because
- 9 times m-reduction [i] would yield (21, 23, large)-net in base 32, but